Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Study Guide and Review - Chapter 8 Use the Distributive Property to factor each polynomial. 35. 12x + 24y SOLUTION: Factor The greatest common factor of each term is 2· 3 or 12. 12x + 24y = 12(x + 2y) 2 2 36. 14x y − 21xy + 35xy SOLUTION: Factor The greatest common factor of each term is 7xy. 2 2 14x y − 21xy + 35xy = 7xy(2x − 3 + 5y) 3 37. 8xy − 16x y + 10y SOLUTION: Factor. The greatest common factor of each term is 2y. 3 3 8xy − 16x y + 10y = 2y(4x − 8x + 5) 2 38. a − 4ac + ab − 4bc SOLUTION: Factor by grouping. 2 39. 2x − 3xz − 2xy + 3yz SOLUTION: 40. 24am − 9an + 40bm − 15bn eSolutions Manual - Powered by Cognero SOLUTION: Page 1 Study Guide and Review - Chapter 8 40. 24am − 9an + 40bm − 15bn SOLUTION: Solve each equation. Check your solutions. 2 41. 6x = 12x SOLUTION: Factor the trinomial using the Zero Product Property. or The roots are 0 and 2. Check by substituting 0 and 2 in for x in the original equation. and The solutions are 0 and 2. 2 42. x = 3x SOLUTION: Factor the trinomial using the Zero Product Property. The roots are 0 and 3. Check by substituting 0 and 3 in for x in the original equation. and eSolutions Manual - Powered by Cognero The solutions are 0 and 3. 2 Page 2 Study Guide and are Review The solutions 0 and-2.Chapter 8 2 42. x = 3x SOLUTION: Factor the trinomial using the Zero Product Property. The roots are 0 and 3. Check by substituting 0 and 3 in for x in the original equation. and The solutions are 0 and 3. 2 43. 3x = 5x SOLUTION: Factor the trinomial using the Zero Product Property. The roots are 0 and . Check by substituting 0 and in for x in the original equation. and The solutions are 0 and . eSolutions Manual - Powered by Cognero 44. x(3x − 6) = 0 SOLUTION: Page 3 Study Guide and Review - Chapter 8 The solutions are 0 and 3. 2 43. 3x = 5x SOLUTION: Factor the trinomial using the Zero Product Property. The roots are 0 and . Check by substituting 0 and in for x in the original equation. and The solutions are 0 and . 44. x(3x − 6) = 0 SOLUTION: Factor the trinomial using the Zero Product Property. x(3x − 6) = 0 x = 0 or The roots are 0 and 2. Check by substituting 0 and 2 in for x in the original equation. and eSolutions Manual - Powered by Cognero The solutions are 0 and 2. Page 4 Study Guide and are Review 8 The solutions 0 and- Chapter . 44. x(3x − 6) = 0 SOLUTION: Factor the trinomial using the Zero Product Property. x(3x − 6) = 0 x = 0 or The roots are 0 and 2. Check by substituting 0 and 2 in for x in the original equation. and The solutions are 0 and 2. 3 2 45. GEOMETRY The area of the rectangle shown is x − 2x + 5x square units. What is the length? SOLUTION: 3 2 2 The area of the rectangle is x − 2x + 5x or x(x – 2x + 5). Area is found by multiplying the length by the width. 2 Because the width is x, the length must be x − 2x + 5. Factor each trinomial. Confirm your answers using a graphing calculator. 2 46. x − 8x + 15 SOLUTION: In this trinomial, b = –8 and c = 15, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 15, and look for the pair of factors with a sum of –8. Factors of 15 –1, –15 –3, –5 Sum of –8 –16 –8 The correct factors are –3 and –5. Check using a Graphing calculator. eSolutions Manual - Powered by Cognero Page 5 SOLUTION: 3 2 2 The area of theReview rectangle is x − 2x8 + 5x or x(x – 2x + 5). Area is found by multiplying the length by the width. Study Guide and - Chapter 2 Because the width is x, the length must be x − 2x + 5. Factor each trinomial. Confirm your answers using a graphing calculator. 2 46. x − 8x + 15 SOLUTION: In this trinomial, b = –8 and c = 15, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 15, and look for the pair of factors with a sum of –8. Factors of 15 –1, –15 –3, –5 Sum of –8 –16 –8 The correct factors are –3 and –5. Check using a Graphing calculator. [– 10, 10] scl: 1 by [– 10, 10] scl: 1 2 47. x + 9x + 20 SOLUTION: In this trinomial, b = 9 and c = 20, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the positive factors of 20, and look for the pair of factors with a sum of 9. Factors of 20 1, 20 2, 10 4, 5 Sum of 9 21 12 9 The correct factors are 4 and 5. Check using a Graphing calculator. eSolutions Manual - Powered by Cognero [– 10, 10] scl: 1 by [– 10, 10] scl: 1 2 Page 6 Study Guide - Chapter [– 10, 10]and scl: Review 1 by [– 10, 10] scl: 18 2 47. x + 9x + 20 SOLUTION: In this trinomial, b = 9 and c = 20, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the positive factors of 20, and look for the pair of factors with a sum of 9. Factors of 20 1, 20 2, 10 4, 5 Sum of 9 21 12 9 The correct factors are 4 and 5. Check using a Graphing calculator. [– 10, 10] scl: 1 by [– 10, 10] scl: 1 2 48. x − 5x − 6 SOLUTION: In this trinomial, b = –5 and c = –6, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of –6, and look for the pair of factors with a sum of –5. Factors of –6 –1, 6 1, –6 2, –3 –2, 3 Sum of –5 5 –5 –1 1 The correct factors are 1 and −6. Check using a Graphing calculator. eSolutions Manual - Powered by Cognero [– 10, 10] scl: 1 by [– 12, 8] scl: 1 Page 7 Study Guide and Review - Chapter 8 [– 10, 10] scl: 1 by [– 10, 10] scl: 1 2 48. x − 5x − 6 SOLUTION: In this trinomial, b = –5 and c = –6, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of –6, and look for the pair of factors with a sum of –5. Factors of –6 –1, 6 1, –6 2, –3 –2, 3 Sum of –5 5 –5 –1 1 The correct factors are 1 and −6. Check using a Graphing calculator. [– 10, 10] scl: 1 by [– 12, 8] scl: 1 2 49. x + 3x − 18 SOLUTION: In this trinomial, b = 3 and c = –18, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of –18, and look for the pair of factors with a sum of 3. Factors of –18 –1, 18 1, –18 –2, 9 2, –9 –3, 6 3, –6 Sum of 3 17 –17 7 –7 3 –3 The correct factors are –3 and 6. Check using a Graphing calculator. eSolutions Manual - Powered by Cognero Page 8 Study Guide and Review - Chapter 8 [– 10, 10] scl: 1 by [– 12, 8] scl: 1 2 49. x + 3x − 18 SOLUTION: In this trinomial, b = 3 and c = –18, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of –18, and look for the pair of factors with a sum of 3. Factors of –18 –1, 18 1, –18 –2, 9 2, –9 –3, 6 3, –6 Sum of 3 17 –17 7 –7 3 –3 The correct factors are –3 and 6. Check using a Graphing calculator. [– 10, 10] scl: 1 by [– 14, 6] scl: 1 Solve each equation. Check your solutions. 2 50. x + 5x − 50 = 0 SOLUTION: The roots are –10 and 5. Check by substituting –10 and 5 in for x in the original equation. and eSolutions Manual - Powered by Cognero The solutions are –10 and 5. Page 9 Study Guide and Review - Chapter 8 [– 10, 10] scl: 1 by [– 14, 6] scl: 1 Solve each equation. Check your solutions. 2 50. x + 5x − 50 = 0 SOLUTION: The roots are –10 and 5. Check by substituting –10 and 5 in for x in the original equation. and The solutions are –10 and 5. 2 51. x − 6x + 8 = 0 SOLUTION: The roots are 2 and 4. Check by substituting 2 and 4 in for x in the original equation. and The solutions are 2 and 4. 2 52. x + 12x + 32 = 0 SOLUTION: eSolutions Manual - Powered by Cognero Page 10 Study Guide and Review - Chapter 8 The solutions are 2 and 4. 2 52. x + 12x + 32 = 0 SOLUTION: The roots are –8 and –4. Check by substituting –8 and –4 in for x in the original equation. and The solutions are –8 and –4. 2 53. x − 2x − 48 = 0 SOLUTION: The roots are –6 and 8. Check by substituting –6 and 8 in for x in the original equation. and The solutions are –6 and 8. 2 54. x + 11x + 10 = 0 SOLUTION: eSolutions Manual - Powered by Cognero Page 11 Study Guide and Review - Chapter 8 The solutions are –6 and 8. 2 54. x + 11x + 10 = 0 SOLUTION: The roots are –10 and –1. Check by substituting –10 and –1 in for x in the original equation. and The solutions are –10 and –1. 55. ART An artist is working on a painting that is 3 inches longer than it is wide. The area of the painting is 154 square inches. What is the length of the painting? SOLUTION: Let x = the width of the painting. Then, x + 3 = the length of the painting. Because a painting cannot have a negative dimension, the width is 11 inches and the length is 11 + 3, or 14 inches. Factor each trinomial, if possible. If the trinomial cannot be factored, write prime . 2 56. 12x + 22x − 14 SOLUTION: In this trinomial, a = 12, b = 22 and c = –14, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 12(–14) or –168 and identify the factors with a sum of 22. Factors of –168 Sum 82 –2, 84 2, –84 –82 53 –3, 56 3, –56 –53 –4,- 42 eSolutions Manual Powered by Cognero 38 Page 12 4, –42 –38 22 –6, 28 Study Guide and Review - Chapter 8 Because a painting cannot have a negative dimension, the width is 11 inches and the length is 11 + 3, or 14 inches. Factor each trinomial, if possible. If the trinomial cannot be factored, write prime . 2 56. 12x + 22x − 14 SOLUTION: In this trinomial, a = 12, b = 22 and c = –14, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 12(–14) or –168 and identify the factors with a sum of 22. Factors of –168 Sum 82 –2, 84 2, –84 –82 53 –3, 56 3, –56 –53 38 –4, 42 4, –42 –38 22 –6, 28 The correct factors are –6 and 28. 2 So, 12x + 22x − 14 = 2(2x − 1)(3x + 7). 2 57. 2y − 9y + 3 SOLUTION: In this trinomial, a = 2, b = –9 and c = 3, so m + p is negative and mp is positive. Therefore, m and p must both be negative. 2(3) = 6 There are no factors of 6 with a sum of –9. So, this trinomial is prime. 2 58. 3x − 6x − 45 SOLUTION: In this trinomial, a = 3, b = –6 and c = –45, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 3(–45) or –135 and identify the factors with a sum of –6. Factors of –135 Sum 134 –1, 135 1, –135 –134 42 –3, 45 3, –45 –42 22 –5, 27 5, –27 –22 6 –9, 15 9, –15 –6 The correct factors are –15 and 9. eSolutions Manual - Powered by Cognero Page 13 negative. 2(3) = 6 Guide and Review - Chapter 8 Study There are no factors of 6 with a sum of –9. So, this trinomial is prime. 2 58. 3x − 6x − 45 SOLUTION: In this trinomial, a = 3, b = –6 and c = –45, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 3(–45) or –135 and identify the factors with a sum of –6. Factors of –135 Sum 134 –1, 135 1, –135 –134 42 –3, 45 3, –45 –42 22 –5, 27 5, –27 –22 6 –9, 15 9, –15 –6 The correct factors are –15 and 9. 2 So, 3x − 6x − 45 = 3(x − 5)(x + 3). 2 59. 2a + 13a − 24 SOLUTION: In this trinomial, a = 2, b = 13 and c = –24, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 2(–24) or –48 and identify the factors with a sum of 13. Factors of –48 Sum 47 –1, 48 1, –48 –47 22 –2, 24 2, –24 –22 13 –3, 16 3, –16 –13 8 –4, 12 4, –12 –8 2 –6, 8 6, –8 –2 The correct factors are –3 and 16. 2 So, 2a + 13a − 24 = (2a − 3)(a + 8). eSolutions Manual - Powered by Cognero Solve each equation. Confirm your answers using a graphing calculator. 2 60. 40x + 2x = 24 Page 14 Study Guide and Review - Chapter 8 2 So, 3x − 6x − 45 = 3(x − 5)(x + 3). 2 59. 2a + 13a − 24 SOLUTION: In this trinomial, a = 2, b = 13 and c = –24, so m + p is positive and mp is negative. Therefore, m and p must have different signs. List the factors of 2(–24) or –48 and identify the factors with a sum of 13. Factors of –48 Sum 47 –1, 48 1, –48 –47 22 –2, 24 2, –24 –22 13 –3, 16 3, –16 –13 8 –4, 12 4, –12 –8 2 –6, 8 6, –8 –2 The correct factors are –3 and 16. 2 So, 2a + 13a − 24 = (2a − 3)(a + 8). Solve each equation. Confirm your answers using a graphing calculator. 2 60. 40x + 2x = 24 SOLUTION: The roots are and or –0.80 and 0.75 . 2 Confirm the roots using a graphing calculator. Let Y1 = 40x + 2x and Y2 = –24. Use the intersect option from the CALC menu to find the points of intersection. eSolutions Manual - Powered by Cognero Page 15 Study Guide and Review - Chapter 8 2 So, 2a + 13a − 24 = (2a − 3)(a + 8). Solve each equation. Confirm your answers using a graphing calculator. 2 60. 40x + 2x = 24 SOLUTION: The roots are and or –0.80 and 0.75 . 2 Confirm the roots using a graphing calculator. Let Y1 = 40x + 2x and Y2 = –24. Use the intersect option from the CALC menu to find the points of intersection. [–5, 5] scl: 1 by [–5, 25] scl: 3 The solutions are and [–5, 5] scl: 1 by [–5, 25] scl: 3 . 2 61. 2x − 3x − 20 = 0 SOLUTION: eSolutions - Powered or by −2.5 and 4. Cognero The Manual roots are Page 16 2 Confirm the roots using a graphing calculator. Let Y1 = 2x − 3x − 20 and Y2 = 0. Use the intersect option from [–5, 5] scl: 1 by [–5, 25] scl: 3 [–5, 5] scl: 1 by [–5, 25] scl: 3 The solutions . Study Guide and are Review and - Chapter 8 2 61. 2x − 3x − 20 = 0 SOLUTION: The roots are or −2.5 and 4. 2 Confirm the roots using a graphing calculator. Let Y1 = 2x − 3x − 20 and Y2 = 0. Use the intersect option from the CALC menu to find the points of intersection. [–10, 10] scl: 1 by [–15, 5] scl: 1 The solutions are [–10, 10] scl: 1 by [–15, 5] scl: 1 and 4. eSolutions Manual - Powered by Cognero Page 17